Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Cumulative distribution function ▷ Cumulative distribution function of discrete random variable

Theorem: Let $X$ be a discrete random variable with possible values $\mathcal{X}$ and probability mass function $f_X(x)$. Then, the cumulative distribution function of $X$ is

\[\label{eq:cdf-pmf} F_X(x) = \sum_{\overset{t \in \mathcal{X}}{t \leq x}} f_X(t) \; .\]

Proof: The cumulative distribution function of a random variable $X$ is defined as the probability that $X$ is smaller than $x$:

\[\label{eq:cdf} F_X(x) = \mathrm{Pr}(X \leq x) \; .\]

The probability mass function of a discrete random variable $X$ returns the probability that $X$ takes a particular value $x$:

\[\label{eq:pmf} f_X(x) = \mathrm{Pr}(X = x) \; .\]

Taking these two definitions together, we have:

\[\label{eq:cdf-pmf-qed} \begin{split} F_X(x) &\overset{\eqref{eq:cdf}}{=} \sum_{\overset{t \in \mathcal{X}}{t \leq x}} \mathrm{Pr}(X = t) \\ &\overset{\eqref{eq:pmf}}{=} \sum_{\overset{t \in \mathcal{X}}{t \leq x}} f_X(t) \; . \end{split}\]
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Metadata: ID: P189 | shortcut: cdf-pmf | author: JoramSoch | date: 2020-11-12, 06:03.